Content uploaded by jean-louis Foulley

Author content

All content in this area was uploaded by jean-louis Foulley on Dec 20, 2013

Content may be subject to copyright.

A preview of the PDF is not available

Mixed Model Methodology has received considerable practical interest over the last two decades. This is due primarily to the following two features: a) MM are tools of choice for analyzing correlated data in a broad area of situations (block x treatment designs; clusters, longitudinal and spatial patterns); b) MM are also increasingly feasible through more and more efficient algorithms (e.g., EM, Average information) and softwares (e.g SAS Proc Mixed, ASReml, R-lme4, Monolix, Nlme, Winbugs, etc...).
We are now in a new stage involving more sophisticated modelling approaches eg multi-level modelling (population and individual; mean and variance models) and also the use of dynamic systems for functional data.
The object of thist text is to present a synthetic approach of the basic theory underlying linear mixed models. At a time when computers and software are easily available and applicable, the purpose of this text is to provide the reader with the elements of a better understanding and mastering of concepts he will use in practice. This is particularly important as most of the existing literature is directed to specific audiences (software -oriented), topics (eg longitudinal data) or methods (maximum likelihood, bayesian approaches).
This course is intended for a broad audience of graduate students, researchers and scientists who are seeking to learn about the foundations of mixed models starting with the linear ones, modelling options and how to apply them in the field of biology, medicine, pharmacology, genetics, genomics, agriculture.

Figures - uploaded by jean-louis Foulley

Author content

All figure content in this area was uploaded by jean-louis Foulley

Content may be subject to copyright.

Content uploaded by jean-louis Foulley

Author content

All content in this area was uploaded by jean-louis Foulley on Dec 20, 2013

Content may be subject to copyright.

A preview of the PDF is not available

... According to the equation of prediction of random effects in (Foulley, 2003), the equation of the predictions of the genetic additive effects can be written ...

Un des principaux enjeux de l’amélioration des plantes consiste aujourd’hui à faire face au changement climatique, en assurant un rendement élevé et plus stable dans des systèmes agricoles économes en intrants (eau, fertilisants) et respectueux de l’environnement. Les nouvelles variétés de blé devront non seulement être tolérantes aux stress hydriques et aux fortes températures, mais aussi continuer à être productives avec des apports limités en fertilisation, tout en maintenant une qualité du grain adaptés aux différents usages. De nouvelles méthodes de prédiction des réponses des blés à ces stress sont indispensables pour avancer dans cette direction.Dans ce travail, nous avons tout d’abord identifié les stress qui régissaient les interactions entre génotypes et les environnements (GxE) dans les essais considérés, puis développé un modèle génomique de l’adaptation à un stress environnemental (Factorial Regression genomic Best Linear Unbiased Prediction ou FR-gBLUP), en particulier pour le stress hydrique. En émettant l’hypothèse que plus des variétés de blés sont génétiquement proches, plus elles répondront de façon similaire à un stress environnemental donné, nous avons mesuré par validation croisée des gains de précision de prédiction par rapport à un modèle additif variant entre 3.5% et 15.4%. Des simulations complètent l’étude en démontrant que plus la part de variance expliquée par les réponses au stress considéré est importante, plus le modèle FR-gBLUP apporte un gain de précision. Pour prédire les réponses variétales à un stress particulier, les environnements doivent être finement caractérisés pour les stress limitant le développement des plantes. En nous intéressant plus particulièrement au stress azoté en France, nous avons établi des indicateurs de stress à partir d’un modèle de culture, et les avons comparés à des indicateurs classiques, tels que le type de conduite azotée ou l’azote disponible. Nous avons ainsi mis en évidence l’intérêt des modèles de culture pour caractériser les interactions GxE et pour prédire la réponse génomique au stress azoté, à condition que le signal d’interaction soit assez fort.Au-delà de l’application potentielle de ces méthodes pour la sélection ou la recommandation de variétés de blés plus adaptées ou plus résistantes au changement climatique, les résultats de ce travail démontrent aussi l’intérêt de la complémentarité des approches éco-physiologiques et génétiques.

In many problems of maximum likelihood estimation, it is impossible to carry out either the E‐step or the M‐step of the EM algorithm. The present paper introduces a gradient algorithm that is closely related to the EM algorithm. This EM gradient algorithm approximately solves the M‐step of the EM algorithm by one iteration of Newton's method. Since Newton's method converges quickly, the local properties of the EM gradient algorithm are almost identical with those of the EM algorithm. Any strict local maximum point of the observed likelihood locally attracts the EM and EM gradient algorithm at the same rate of convergence, and near the maximum point the EM gradient algorithm always produces an increase in the likelihood. With proper modification the EM gradient algorithm also exhibits global convergence properties that are similar to those of the EM algorithm. Our proof of global convergence applies and improves existing theory for the EM algorithm. These theoretical points are reinforced by a discussion of three realistic examples illustrating how the EM gradient algorithm can succeed where the EM algorithm is intractable.

Patterson and Thompson (1971) proposed estimating the variance components of a mixed analysis of variance model by maximizing the likelihood of a set of error contrasts. A convenient representation is obtained for that likelihood. It is shown that, from a Bayesian viewpoint, using only error contrasts to make inferences on variance components is equivalent to ignoring any prior information on the fixed effects and using all the data.